Fixed point of closed disk

Question

Let $D = \{(x,y)\in \mathbb{R}^2: x^2 + y^2 ≤ 1 \}$. Let $A \subset \mathrm{int}D$. Let $A$ be connected and compact and let $D \setminus A$ be connected. Let $f:A \longrightarrow A$ be a continuous function and let $g:D \longrightarrow D$ be also continuous function such that $g_{|A} = f$.$\\$ Does it imply that there exists $x \in \mathrm{int}D$ such that $g(x)=x$?